Solve the differential equation (2x + y + 3)dx = (2y + x + 1)dy.
Sol: The given equation can be written as
------ (i)
Here a = 2, b = 1, a' = 1, b' = 2
Hence, b ≠ – a' and a/a' = b/b'
Therefore, the given equation can be solved by case (iii).
Substitute x = X + h, y = Y + k in (i) and differentiating w.r.t. x
------- (ii)
Now choose h and k such that
2h + k + 3 = 0
and h + 2k + 1 = 0
Solving them for h and k, we get
Therefore, (iii) becomes
Integrating both sides, we have
– (3/2) log(1 – V) – (1/2) log(1 + V) = 2 log X – log c
3 log(1 – V) + log(1 + V) + 4log X = 2 log c
log[(1 – V)3 (1 + V)X4] = log c2
X4(1 – V)3 (1 + V) = c2